1. No category

Math 3220 § 1.
Treibergs σ−
ιι
Sample Problems for the Third Midterm Exam
Name: Problems With Solutions
November 2, 2007
Questions 1–4 appeared in my Fall 2000 and Fall 2001 Math 3220 exams.
(1) Let f : Rn → Rn be differentiable at a ∈ Rn .
(a.) Let g : Rn → R be defined by g(x) = x • f (x), (dot product.) Find the total derivative
(differential) Dg(a)(h) where h ∈ Rn .
(b.) Without using the product theorem, prove your answer.
(2) For each part, determine whether the statement is TRUE or FALSE. If the statement is true, give a
justification. If the statement is false, give a counterexample. You may use theorems.
(a.) Let f : R2 → R be continuous. Suppose that for all (x, y) ∈ R2 both
lim
h→0
f (x + h, y) − f (x, y)
h
and
lim
k→0
f (x, y + k) − f (x, y)
k
exist. Then f is differentiable at (1, 2).
(b.) Suppose f : R2 → R is a C 2 function for which the third partial derivatives fxxy (x, y) exist for
all (x, y) ∈ R2 such that fxxy (x, y) is continuous at (0, 0). Then fxyx (0, 0) and fyxx (0, 0) exist and are
equal fxxy (0, 0) = fxyx (0, 0) = fyxx (0, 0).
(3) Suppose f , g : R2 → R2 . Assume that g is differentiable at x0 ∈ R2 and that for some α > 1 and
M < ∞ we have
kf (x) − g(x)k ≤ M kx − x0 kα
for all x ∈ R2 . Show that f is differentiable at x0 and that Df (x0 ) = Dg(x0 ).
(4) Let F : R5 → R2 be given by F = (f1 , f2 ) where
f1 (v, w, x, y, z) = v + w2 + x + y,
f2 (v, w, x, y, z) = vy + wz.
Suppose that there is an open neighborhood U ⊂ R3 containing the point (3, 4, 5) and C 1 functions
G : U → R2 where G = (g1 , g2 ) so that for all (x, y, z) ∈ U ,
g1 (3, 4, 5) =1,
f1 (g1 (x, y, z), g2 (x, y, z), x, y, z) =12,
g2 (3, 4, 5) =2;
f2 (g1 (x, y, z), g2 (x, y, z), x, y, z) =14.
What is the differential DG(3, 4, 5)(h, k, `) ?
E1. Suppose f : R3 → R2 is given below. Is f differentiable on R3 ? If so, find the differential
Df (x, y, z)(h, k, `).
 
x
xy + x2 z 3


f y =
.
x4 + y + y 5 z 6
z
E2. Find the extrema of ϕ(x, y, z) = x2 + y 2 + z 2 subject to the constraints x − y = 1 and y 2 − z 2 = 1.
¯ ⊆ Br (0), and that there exists
E3. Suppose that r and α are positive, E ⊆ Rn is a convex set such that E
a sequence xk ∈ E such that xk → 0 as k → ∞. If f : Br (0) → R is continuously differentiable and
|f (x)| ≤ kxkα for all x ∈ E, prove that there is an M < ∞ such that |f (x)| ≤ M kxk for x ∈ E.
E4. Show that f (x, y) has partial derivatives for all (x, y) ∈ R2 but f is not differentiable at (0, 0), where
 3
3
2
 x + 2y + 4xy , if (x, y) 6= (0, 0);
x2 + 2y 2
f (x, y) =

0,
if (x, y) = (0, 0).
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E5. Suppose that f : Rp → R is differentiable at a in Rp and df (a) 6= 0. Show that ∇f (a) points in the
direction of fastest increase (that has the largest directional derivative.)
E6. Show that S ⊆ R4 is locally a parameterized 2-surface near b = (0, 2, 0, 2), where
S = {(x, y, z, w) ∈ R4 : x2 + y 2 + z 2 + w2 = 8, x + y + z − w = 0}.
Find the tangent plane to S at b.
E7. Suppose that f : Rp → R is C 3 and some point a ∈ Rp is critical df (a) = 0 and d2 f (a) has both a
positive and a negative eigenvalue. (Since second derivatives are continuous, d2 f (a) is symmetric thus
all of its eigenvalues are real.) Show that the critical point a is a saddle point: for every δ > 0 there are
points x, y ∈ Bδ (a) so that f (x) < f (a) < f (y).
E8. Let F : R3 → R3 be given by F (x, y, z) = (x2 + y 2 , xz, y 3 − z 3 ). Assume that there is an open set U
about P0 = (3, 1, 2) and that V = F (U ) is an open set about Q0 = F (P0 ) on which F has an inverse
function F −1 ∈ C 1 (V, U ). Find D[F −1 ](Q0 ).
E9. Suppose G : R5 → R3 is given by G(p, q, x, y, z) = (px + y 2 , q 2 z, py − qz + x). Assume that there is an
open set U around (3, 2) and a C 1 function H : U → R3 so that H(3, 2) = (1, 5, 4) and for all (p, q) ∈ U
we have G(p, q, H(p, q)) = (28, 16, 8). Find DH(3, 2).
Solutions.
(1.) Let f : Rn → Rn be differentiable at a ∈ Rn . Let g : Rn → R be defined by g(x) = x • f (x), (dot
product.) Find the total derivative (differential) Dg(a)(h) where h ∈ Rn .
The product rule gives Dg(x)(h) = D(x • f (x))(h) = h • f (x) + x • Df (x)(h). This is the differential because
k(x + h) • f (x + h) − x • f (x) − h • f (x) − x • Df (x)(h)k
kg(x + h) − g(x) − Dg(x)(h)k
= lim
h→0
h→0
khk
khk
kx • (f (x + h) − f (x) − Df (x)(h)) + h • (f (x + h) − f (x))k
= lim
h→0
khk
kf (x + h) − f (x) − Df (x)(h)k khk
≤ lim kxk
+
kf (x + h) − f (x)k = kxk · 0 + 1 · 0 = 0
h→0
khk
khk
lim
using f is continuous at a which follows from f being differentiable at a.
(2a.) Let f : R2 → R be continuous. Suppose that for all (x, y) ∈ R2 both
lim
h→0
f (x + h, y) − f (x, y)
h
and
lim
k→0
f (x, y + k) − f (x, y)
k
exist. Then f is differentiable at (1, 2).
FALSE! The two limits are nothing more than fx (x, y) and fy (x, y), the partial derivatives. There are functions
where the partial derivatives exist at all points, but the function is not differentiable. (If it were known that fx
and fy are continuous at some point a ∈ R2 , then our theorem says that the function would be differentiable
at a.) An example of such a function is
(
f (x, y) =
(x−1)2 (y−2)
(x−1)2 +(y−2)2 ,
if (x, y) 6= (1, 2);
0.
if (x, y) = (1, 2).
Away from (1, 2), the denominator avoids zero, so the partial derivatives exist and are continuous, hence f is
differentiable. Also, observe that f (1, y) = f (x, 2) = 0 for all x, y. Hence fy (1, y) = fx (x, 2) = 0 so the partial
derivatives exist at (1, 2). If the function were differentiable at (1, 2), then the vanishing of the partial derivatives
implies that the differential would have to be T (h, k) = 0 all h, k. But the limit
kf (1 + h, 2 + k) − f (1, 2) − T (h, k)k
kf (1 + h, 2 + k)k
=
lim
k(h, k)k
k(h, k)k
(h,k)→(1,2)
(h,k)→(1,2)
lim
3. Math 3220
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Third Midterm Exam
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does not exist. To see this, consider the first approach (h, k) = (t, 0) as t → 0. The numerator vanishes so
along this approach the limit would be √
zero. Then consider the second approach (h, k) = (t,
√ t) for t > 0. Then
f (1 + t, 2 + t) = t/2 and k(t, t)k = 2|t|. Then the difference quotient tends to 1/(2 2). Since the two
approaches have different limits, there is no two dimensional limit: the function is not differentiable at (1, 2).
(2b.) Suppose f : R2 → R is a C 2 function for which the third partial derivatives fxxy (x, y) exist for all
(x, y) ∈ R2 such that fxxy (x, y) is continuous at (0, 0). Then fxyx (0, 0) and fyxx (0, 0) exist and are equal
fxxy (0, 0) = fxyx (0, 0) = fyxx (0, 0).
TRUE! This is just an application of the equality of cross partials theorem to fx and fy which are C 1 by
assumption, since f is C 2 . We are given that (fx )xy exists and is continuous at (0, 0). But, this is sufficient
to be able to assert the existence of the other mixed partial derivative, and that it is equal to the first at the
point (fx )xy (0, 0) = (fx )yx (0, 0). But since f ∈ C 2 (R2 ), we also have that fxy = fyx for all of R2 . Hence,
all third derivatives exist and are equal fxxy (0, 0) = (fx )xy (0, 0) = (fx )yx (0, 0) = (fxy )x (0, 0) = (fyx )x (0, 0) =
fyxx (0, 0).
(3.) Suppose f , g : R2 → R2 . Assume that g is differentiable at x0 ∈ R2 and that for some α > 1 and M < ∞
we have
kf (x) − g(x)k ≤ M kx − x0 kα
for all x ∈ R2 . Show that f is differentiable at x0 and that Df (x0 ) = Dg(x0 ).
It suffices to show that the difference quotient limits to zero. Now,
kf (x + h) − f (x) − Dg(x)(h)k
=
khk
kf (x + h) − g(x + h) − f (x) + g(x) + g(x + h) − g(x) − Dg(x)(h)k
= lim
h→0
khk
kf (x + h) − g(x + h)k + kg(x) − f (x)k + kg(x + h) − g(x) − Dg(x)(h)k
≤ lim
h→0
khk
M khkα + M k0kα
kg(x + h) − g(x) − Dg(x)(h)k
≤ lim
+
h→0
khk
khk
kg(x
+
h)
−
g(x)
−
Dg(x)(h)k
= 0 + 0.
= lim M khkα−1 + lim
h→0
h→0
khk
lim
h→0
(4*.) (Slight generalization.) Let F : R5 → R2 be given by F = (f1 , f2 ) where f1 (v, w, x, y, z) = v +w2 +x+y,
f2 (v, w, x, y, z) = vy + wz. Show that there is a neighborhood U ⊂ R3 containing the point (3, 4, 5) and C 1
functions G : U → R2 where G = (g1 , g2 ) so that g1 (3, 4, 5) = 1, g2 (3, 4, 5) = 2 and for all (x, y, z) ∈ U ,
f1 (g1 (x, y, z), g2 (x, y, z), x, y, z) = 12, f2 (g1 (x, y, z), g2 (x, y, z), x, y, z) = 14. Find the total derivative
DG(3, 4, 5)(h, j, k).
You were given the first conclusion in problem (4.) which can be answered knowing the chain rule. (4*) is
an application of the Implicit Function Theorem. (We have not talked about this in class yet, The theorem says
that if there is enough differentiability, and if the problem can be solved for the linear approximations given by
the differential, then, at least in a small neighborhood, the nonlinear problem can be solved as well. You will
not be asked to apply the Implicit Function Theorem, but you should be able to solve such problems, assuming
its conclusions.) The function F : R2+3 → R2 is C 1 on R5 such that F (1, 2, 3, 4, 5) = (12, 14) = c. To
solve for v and w in terms of (x, y, z) we need to be able to solve the linearization. If we put u = (v, w) and
x = (x, y, z), we are looking for G : R3 → R2 so that F (G(x), x) = c and G(3, 4, 5) = (1, 2). Taking Dx
gives Du F (G(x), x) ◦ DG(x) + Dx F (G(x), x) = 0 which says that we may solve for the differential DG(x)
whenever Du F (G(x), x) is invertible and then DG(x) = −[Du F (G(x), x)]−1 ◦ Dx F (G(x), x). At the center
point (3, 4, 5), the matrix of the transformation is
∂f1 ∂f1 1 2w 1 4
∂v
∂w
Du F (G(3, 4, 5), (3, 4, 5)) = ∂f
=
=
∂f2
2
y z (u,x)=(1,2,3,4,5)
4 5
x=(3,4,5)
∂v
∂w
which is invertible. Hence there is an open set U ⊂ R3 such that (3, 4, 5) ∈ U and a C 1 function G : U → R2
so that G(2, 3, 4) = (1, 2) and F (G(x), x) = c for all x ∈ U . (Thus, we have checked the differentiability and
4. Math 3220
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Third Midterm Exam
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the solubility of the linearized problem is satisfied. The IFT gives the existence of a G ∈ C 1 (U, R2 ) so that
G(2, 3, 4) = (1, 2) and F (G(x), x) = c for all x ∈ U . You would be given this as a hypothesis. Then take the
total derivative of F (G(x), x) = c with respect to x and solve for DG(x) at the given point, as above.) By the
formula for the differential
 
 
h
h
DG(3, 4, 5)  k  = −[Du F (G(3, 4, 5), (3, 4, 5))]−1 ◦ Dx F (G(3, 4, 5), (3, 4, 5))  k 
`
`
 
#−1  ∂f
!
"
∂f1
∂f1
 h

1
∂f1
∂f1
∂x
∂y
∂z
∂v
∂w
k
=−
∂f2
∂f2
∂f2
∂f2
∂f2


x=(3,4,5)
∂v
∂w
∂x
∂y
∂z
`
x=(3,4,5)
"
#−1 ( ) 
h
1 2w 1 1 0 k
=−
y z (u,x)=(1,2,3,4,5)
0 v w (u,x)=(1,2,3,4,5)
`
 
 
−1 h
h
1
1 4
1 1 0  
5 −4
1 1 0  
k =
=−
k
4 5
0 1 2
−4
1
0 1 2
11
`
`
 
h
1 −5h − k + 8`
1 −5 −1 8  
k =
=
. 4
3 −2
11
11 4h + 3k − 2`
`
(E1.) Suppose f : R3 → R2 is given by f (x, y, z) = (xy + x2 z 3 , x4 + y + y 5 z 6 ). Is f differentiable on R3 ? If
so, find the differential df (x, y, z)(h, k, `).
YES! The partial derivatives are
 
 
 
2 2
x
x
x
3
∂f  
∂f  
∂f  
y + 2xz
x
3x z
,
,
=
=
y =
y
y
4x3
1 + 5y 4 z 6
6y 5 z 5
∂x
∂y
∂z
z
z
z
Since f is a polynomial function, its first partial derivatives exist at all points and are polynomial functions. But
by the theorem giving conditions for differentiability, since the partial derivatives are continuous at all points, the
function is differentiable at all points. The differential is given by the 2 × 3 Jacobian matrix
  
 
!  h
∂f1
∂f1
∂f1
h
x
h
3
2 2
y
+
2xz
x
3x
z
∂x
∂y
∂z
k =
k.
df  y   k  = ∂f
3
4
5 5
∂f2
∂f2
2
4x
1
+
5y
z66
6y
z
∂x
∂y
∂z
`
z
`
`
(E2.) Find the extrema of ϕ(x, y, z) = x2 + y 2 + z 2 subject to the constraints x − y = 1 and y 2 − z 2 = 1.
Let g(x, y, z) = x − y − 1 and h(x, y, z) = y 2 − z 2 − 1. Then using Lagrange Multipliers, the extrema occur
as solutions (x, y, z, λ, µ) of the system g = 0, h = 0, ∇ϕ = λ∇g + µ∇h. Hence
(2x, 2y, 2z) = λ(1, −1, 0) + µ(0, 2y, −2z).
Thus 2x = λ, 2y = −λ + 2µy and z = −µz. From the last equation, either z = 0 or µ = −1.
If µ = −1 then λ = −4y. Since 2x = λ we get x + 2y = 0. Now g = 0 implies y = 1/3. However, h = 0
implies 0 = 1/9 − z 2 − 1 which is a contradiction.
If z = 0 then h = 0 implies y = ±1. Since g = 0 we have x = 2 when y = 1 and x = 0 when y = −1. Thus
the only candidates for extrema subject to the constraints g = h = 0 are ϕ(2, 1, 0) = 5 and ϕ(0, −1, 0) = 1. To
see whether these are maxima or minima, consider the geometric interpretation. The problem asks to find the
closest or farthest points of the solution set g = h = 0 to the origin. However h = 0 corresponds to a hyperbolic
cylinder (union of lines parallel to the x-axis that pass through a hyperbola in the y-z plane.) The constraint
g = 0 is a plane parallel to the z-axis that crosses the hyperboloid. Thus the constraint set consists of two
5. Math 3220
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hyperbolas in the g = 0 plane. The level curves on ϕ are circles in this plane. As there are only two candidates
for extrema, these occur when the circles meet (are tangent to) the two arcs of the hyperbola. Larger circles cross
the arcs. This means that (0, −1, 0) is a minimum and (2, 1, 0) is a local but not global minimum, and there is
no maximum since one can attain arbitrarily large ϕ on the constraint set.
¯ ⊆ Br (0), and that there exists a
(E3.) Suppose that r and α are positive, E ⊆ Rn is a convex set such that E
sequence xk ∈ E such that xk → 0 as k → ∞. If f : Br (0) → R is continuously differentiable and |f (x)| ≤ kxkα
for all x ∈ E, prove that there is an M < ∞ such that |f (x)| ≤ M kxk for x ∈ E.
¯ is closed and
Since the partial derivatives are continuous, the function x 7→ kdf (x)k is continuous. As E
¯ < ∞. Since
bounded, it is compact so the continuous function attains its maximum M = sup{kdf (x)k : x ∈ E}
E is convex, for any pair of points x, xk ∈ E the line segment [xk , x] is in E. Since f is differentiable in a
neighborhood of [xk , x], we may apply the Mean Value Theorem. There is a c ∈ [xk , x] so that
f (x) = f (xk ) + df (c)(x − xk )
Estimating using triangle and Schwarz inequalities, and the hypothesis
|f (x)| ≤ |f (xk )| + kdf (c)kkx − xk k
≤ kxk kα + M kx − xk k.
Now since α > 0, by passing to the limit as k → ∞ we obtain the estimate
|f (x)| ≤ 0 + M kxk.
(E4.) T heorem. f (x, y) has partial derivatives for all (x, y) ∈ R2 but f is not differentiable at (0, 0), where
 3
3
2
 x + 2y + 4xy , if (x, y) 6= (0, 0);
2
2
x + 2y
f (x, y) =

0,
if (x, y) = (0, 0).
Since f is a rational function whose denominator is nonzero away from (0, 0), the partial derivatives exist and
are rational functions there. At (0, 0),
∂f
f (h, 0) − f (0, 0)
(0, 0) = lim
= lim
h→0
h→0
∂x
h
and
∂f
f (0, h) − f (0, 0)
(0, 0) = lim
= lim
h→0
h→0
∂y
h
h3
h2
−0
=1
h
2h3
2h2
−0
=1
h
so both partial derivatives exist at (0, 0) as well. Therefore, supposing that f were differentiable at (0, 0), its
differential would be df (0, 0) = (1, 1). Now lets check if this differential well-approximates f near zero.
f (0 + h, 0 + k) − f (0, 0) − df (0, 0)
lim
k(h, k) − (0, 0)k
(h,k)→(0,0)
h
k
=
h3 +2k3 +4hk2
−0−
h2 +2k2
√
(h,k)→(0,0)
h2 + k 2
=
2hk 2 − h2 k
√
.
(h,k)→(0,0) (h2 + 2k 2 ) h2 + k 2
lim
(h + k)
lim
√
Along the path (h, k) = (t, 0) the limit is zero. However, taking the path (h, k) = (t, t), the limit is 1/(3 2).
Because the limits along two approach paths disagree, there is no two dimensional limit. The function is not well
approximated by the only possible affine function, hence it is not differentiable at (0, 0).
(E5.) T heorem. Suppose that f : Rp → R is differentiable at a ∈ Rp and df (a) 6= 0. Then ∇f (a) points in
the direction of fastest increase (that has the largest directional derivative.)
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Let u be a unit vector. Then the directional derivative in the u direction is Du f (a) = ∇f (a) · u. By the
Schwarz inequality |Du f (a)| ≤ k∇f (a)kkuk = k∇f (a)k with equality only if u = ±∇f (a)/k∇f (a)k. Thus
Du f (a) = k∇f (a)k if and only if u = ∇f (a)/k∇f (a)k, which is in the gradient direction.
(E6.) T heorem. S ⊆ R4 is locally a parameterized 2-surface near b = (0, 2, 0, 2), where
S = {(x, y, z, w) ∈ R4 : x2 + y 2 + z 2 + w2 = 8, x + y + z − w = 0}.
the tangent plane to S at b is {(s, 2 − 2s − 2t , t, 2 + 2s + 2t ) : s, t ∈ R}.
√
The surface is the intersection of a 3-plane through the origin and the 3-sphere about the origin of radius 2 2
which is a great 2-sphere. Let’s write the surface near b as a graph over the (x, z)-plane, for x2 + z 2 < 1. Solving
the second equation gives y = w − x − z. Substituting x = u and z = v into the first and solving for w gives
G : U → R4 where we take the “+” square root because G(0, 0) = (0, 2, 0, 2) and U = B1 ((0, 0)):
!
√
√
u + v + 16 − 3u2 − 2uv − 3v 2
−u − v + 16 − 3u2 − 2uv − 3v 2
G(u, v) = u,
, v,
2
2
G ∈ C 1 (U, R4 ) because 16 − 3(u2 + v 2 ) − 2uv > 16 − 3 · 1 − 1 = 12 since 2uv ≤ u2 + v 2 < 1. One checks that
G(u, v) ∈ S. Let V = {(x, y, z, w) : x2 + z 2 < 1, y > 0, w > 0}. One checks that b ∈ S ∩ V = G(U ). We need
that G : U → S ∩ V is one-to-one. But if (ui , vi ) ∈ U and G((u1 , v1 )) = G((u2 , v2 )) then x- and z-coordinate
functions give u1 = u2 and v1 = v2 so G is one-to-one. Finally we check that G is two dimensional. dG((u, v))
is a 4 × 2 matrix given by




1
0
1
0
1
1
−3u−v
−u−3v
1
1
 −2 − √
− 2 − 2√16−3u2 −2uv−3v2 
−2 
−
2 16−3u2 −2uv−3v 2
,
dG((u, v)) = 
dG((0, 0)) =  2
.


0
1
0
1
1
−3u−v
−u−3v
1
1
1
√
√
2 − 2 16−3u2 −2uv−3v 2
2 − 2 16−3u2 −2uv−3v 2
2
2
dG is rank two because the first and third rows are independent, hence G(U ) is a parametrized 2-surface. The
tangent space is b + dG((0, 0))(R2 ).
(E7.) T heorem. Suppose that f : Rp → R is C 3 and some point a ∈ Rp is critical df (a) = 0 and d2 f (a) has
both a positive and a negative eigenvalue. Then the critical point a is a saddle point: for every δ > 0 there are
points x, y ∈ Bδ (a) so that f (x) < f (a) < f (y).
By continuity of(the third partial derivatives, a continuous
) function takes its maximum on a compact set,
2
3
P
∂ f (u)
namely M 2 = sup
: ku − ak ≤ 1 . It follows that for any h ∈ Rp , and c ∈ B1 (a)
i,j,k
∂xi ∂xj ∂xk
P
∂ 3 f (c)
that |d3 f (c)(h)3 | = i,j,k
hi hj hk ≤ M khk3 by applying the Schwarz inequality to each sum. Now
∂xi ∂xj ∂xk
by assumption there are v, w ∈ Rp unit eigenvectors such that d2 f (a)v = λ1 v and d2 f (a)w = λ2 w with
λ1 < 0 < λ2 . Assume that 0 < δ < 1 is so small that M δ < min{|λ1 |, |λ2 |}. Since second derivatives are
differentiable, we can apply Taylor’s formula to second order. We get for each 0 < t < δ some c ∈ (0, 1) such
that
1
1
f (a + tv) = f (a) + df (a)(tv) + t2 v · d2 f (a)v + t3 d3 f (a + ctv)(v)3
2
6
1
1 3
2
3
≤ f (a) + 0 + λ1 t v · v + t M kvk
2
6
1 2
1
1 2
1
≤ f (a) + t λ1 + tM ≤ f (a) + t λ1 + |λ1 | < f (a).
2
3
2
3
Similarly, for each 0 < t < δ, there is a c ∈ (0, 1) so that
1
1
1
1
f (a + tw) ≥ f (a) + λ2 t2 w · w − t3 M kwk3 ≥ f (a) − t2 λ2 − tM
2
6
2
3
1 2
1
≥ f (a) + t λ2 − |λ2 | > f (a).
2
3
7. Math 3220
§
1.
Third Midterm Exam
Name:
(E8*.) (Slight generalization.) T heorem. Let F : R3 → R3 be given by F (x, y, z) = (x2 + y 2 , xz, y 3 − z 3 ).
Then there is an open set U about P0 = (3, 1, 2) so that F is invertible on U and that F (U ) is an open set about
Q0 = F (P0 ) on which F −1 is C 1 . Find D[F −1 ](Q0 ). We find D[F −1 ](Q) where Q ∈ F (U ).
(The function F (x, y, z) polynomial, therefore C 1 . We check that the determinant of the Jacobian matrix
∆F (P0 ) 6= 0 and all of the conclusions follow from the Inverse Function Theorem. The fact that the linearization
was invertible at the point enables you to conclude the existence of an inverse function. You were, however,
given this in the problem.) Let G ∈ C 1 (V, R3 ) be the inverse function of F . Thus in U we have the equation
F (G(x, y, z)) = (x, y, z). Apply the chain rule, and solve for DG at the point. Thus D(F ◦ G) = DF (P0 ) ◦
DG(Q0 ) = I so DG(Q0 ) = (DF (P0 ))−1 . The matrix of DF (P0 ) is the Jacobian matrix
 ∂F1

DF (P ) = 
∂x
∂F2
∂x
∂F3
∂x
∂F1
∂y
∂F2
∂y
∂F3
∂y
∂F1
∂z
∂F2
∂z
∂F3
∂z


2x 2y
 
=
z
0

0 3y 2

0
x ;
−3z 2

6
DF (P0 ) =  2
0

2
0
0
3 .
3 −12
Thus ∆F (P0 ) = det(DFx (P0 )) = 6 6= 0. Finally, for Q near (10, 6, −7) = Q0 = F (P0 ), so DF −1 (Q) =

[DF (F −1 (Q))]−1
−3xy 2
1
 6yz 2
=
6yz 3 − 6x2 y 2
2xy
3z 3
−6xz 2
−2x2

3zy 2
−6xy 2  ;
−2yz
3
2

−4 −1
DF −1 (Q0 ) =  −4 12
3 
2
−1 3
3

where (x, y, z) = F −1 (Q). (E9*.) (Slight generalization.) T heorem. Suppose G : R5 → R3 is given by G(p, q, x, y, z) = (px + y 2 , q 2 z, py −
qz + x). Then there is an open set U around T0 = (3, 2) and a C 1 function H : U → R3 so that H(3, 2) =
(1, 5, 4) = X0 and for all (p, q) ∈ U we have G(p, q, H(p, q)) = (28, 16, 8). We find DH(3, 2) and DH(p, q).
The function G is polynomial so C 1 . We have to check that the linearization is soluble at (3, 2, 1, 5, 4). Let
T0 = (3, 2) and X0 = (1, 5, 4). This follows if the Dx G part of the Jacobian matrix is invertible.
 ∂G1

Dx G = 
∂x
∂G2
∂x
∂G3
∂x
∂G1
∂y
∂G2
∂y
∂G3
∂y
∂G1
∂z
∂G2
∂z
∂G3
∂z


p
 
= 0
1
2y
0
p

0
q2  ;
−q

3
Dx G(T0 , X0 ) =  0
1

10 0
0
4 
3 −2
which is invertible since its determinant is 4. The Implicit Function Theorem applies and yields the C 1 function H as
desired. (Again, you were given that there is H ∈ C 1 (U ) satisfying F (p, q, H(p, q)) = (28, 16, 8) for all (p, q) ∈ U .
Find the differential of H by differentiating the equation using the chain rule. Think of H : U → U × R3 is given
by H(p, q) = (p, q, H(p, q)), and then differentiate G ◦ H = const using the chain rule. Thus Dt H = DIt H so
0 = Dt (G ◦ H) = Dt G + Dx G ◦ Dt H. Here the total derivative matrix has columns associated to t = (p, q) and
columns associated to x = (x, y, z) drivatives, DG = (Dt G, Dx G). To find the total derivative of H we need
the other part of the Jacobian
 ∂G1

Dt G = 
∂p
∂G2
∂p
∂G3
∂p
∂G1
∂q
∂G2
∂q
∂G3
∂q



x 0
 
 = 0 2qz  ;
y −z


1 0
Dt G(T0 , X0 ) =  0 16  ;
5 −4

−3
DH(T0 ) =  − 25
4
−10
since total derivative of implicit function DH(T ) = −[Dx G(T, H(T ))]−1 Dt G(T, H(T )) =

pq 2
1 
2yq
∆
2yq 2
q2
−pq
−pq 2



0
x 0
xpq 2
1 
2y − p2   0 2qz  =
2xyq + 2y 2 − yp2
∆
0
y −z
2xyq 2
where ∆ = q 2 (p2 − 2y) and (x, y, z) = H(p, q). 
2q 3 z
p2 z − 2pq 2 z − 2yz 
−2pq 3 z

−16
−25 
48